Let X and Y be two events such that P(X / Y) = \frac{1}{2} , P(Y / X) = \frac{1}{3} and P \left(\right. X \cap Y \left.\right) = \frac{1}{6} . Which of the following is(are) correct?

Engineering

Mathematics

Conditional Probability

Question

Let X and Y be two events such that P(X / Y) = $\frac{1}{2}$ , P(Y / X) = $\frac{1}{3}$ and $P (X \cap Y) = \frac{1}{6}$ . Which of the following is(are) correct?

$P (X^{c} \cap Y) = \frac{1}{3}$

$P (X \cup Y) = \frac{2}{3}$

X and Y are independent

X and Y are not independent

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$P (X / Y) = \frac{1}{2}$

$\frac{P (X \cap Y)}{P (Y)} = \frac{1}{2} \Rightarrow P (Y) = \frac{1}{3}$

$\frac{P (X \cap Y)}{P (X)} = \frac{1}{3} \Rightarrow P (X) = \frac{1}{2}$

P(X $\cap$ Y) = P(X) · P(Y) $\Rightarrow$ X, Y are independent

$P (X \cup Y) = \frac{1}{2} + \frac{1}{3} - \frac{1}{6} = \frac{5}{6} - \frac{1}{6} = \frac{2}{3}$

$P (X^{c} \cup Y) = P (Y) - P (X \cap Y) = \frac{1}{3} - \frac{1}{6} = \frac{1}{6}$